Positive-definite function on a group
In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.
Definition
Let be a group, be a complex Hilbert space, and be the bounded operators on.A positive-definite function on is a function that satisfies
for every function with finite support.
In other words, a function is said to be a positive-definite function if the kernel defined by is a positive-definite kernel. Such a kernel is -symmetric, that is, it invariant under left -action: When is a locally compact group, the definition generalizes by integration over its left-invariant Haar measure. A positive-definite function on is a continuous function that satisfiesfor every continuous function with compact support.
Examples
The constant function, where is the identity operator on, is positive-definite.Let be a finite abelian group and be the one-dimensional Hilbert space. Any character is positive-definite.
To show this, recall that a character of a finite group is a homomorphism from to the multiplicative group of norm-1 complex numbers. Then, for any function, When with the Lebesgue measure, and, a positive-definite function on is a continuous function such thatfor every continuous function with compact support.
Unitary representations
A unitary representation is a unital homomorphism where is a unitary operator for all. For such,.Positive-definite functions on are intimately related to unitary representations of. Every unitary representation of gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of in a natural way.
Let be a unitary representation of. If is the projection onto a closed subspace of. Then is a positive-definite function on with values in. This can be shown readily:
for every with finite support. If has a topology and is weakly continuous, then clearly so is.
On the other hand, consider now a positive-definite function on. A unitary representation of can be obtained as follows. Let be the family of functions with finite support. The corresponding positive kernel defines a inner product on. Let the resulting Hilbert space be denoted by.
We notice that the "matrix elements" for all in. So preserves the inner product on, i.e. it is unitary in. It is clear that the map is a representation of on.
The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:
where denotes the closure of the linear span.
Identify as elements in, whose support consists of the identity element, and let be the projection onto this subspace. Then we have for all.